Partial equilibrium models of externalities assume a given set of prices to
firms that are interacting with production connectedness (Runge 1981, Varian
1994). Runge formulates a continuous model that reduces to a prisoner's
dilemma or an assurance game, depending on the assumptions about cost
functions. This appendix reviews his demonstration of the suboptimality of
private decisions, and demonstrates the optimality of a solution in which the
firms are required to divide their profits among each other.

Runge defines the following problem, Let:

Thus,

An externality arises if:

A social optimum maximizes joint profit, where

Therefore

The first order conditions are

The solution to these two equations in two unknowns gives the optimum level of
production for each of the two goods.

Now examine a market equilibrium where each firm keeps its profits. The
individual firm, for instance firm 1, solves the problem of maximizing its
profits,

and the first order condition is

which will give a different value of q1 than the social
optimum, because the first order condition omits the term involving the partial
derivative of the second firm's cost function.

Now suppose

u = Share of firm 1's profit kept by firm 1

v = Share of firm 2's profit kept by firm 2

Both u and v can hold a value from zero to one.

If firm 1's net return is defined by

Then firm 1's maximization problem is to maximize

The first order condition is

Thus, if u = v = 1/2, the symmetric generosity rule, then firm 1's first
order condition equals the social first order condition.

This example can be generalized to many firms. Let Tj =
return to firm j after application of forced generosity using the notation of
section II above,

Let

Then, the return to firm j is determined as follows:

where

vij = share of Si that is distributed to
firm j.

Examining the maximization problem for firm 1

The first order condition for the optimum choice of q1 is

If a symmetric generosity rule applies, then vij = 1/n and

This condition is identical to the condition for social optimality.

Runge points out that if the cost functions are separable, each firm can solve
its individual maximization problem without knowing what the other firms have
decided. This is true for both the pure private profit case or the symmetric
generosity case. Separability means that each cost function is additive:

as a result,

If the cost functions are not separable, then an individual firm cannot
determine its point of maximum profit without knowing what other firms are
doing in either the pure private profit case or the symmetric generosity case.
Non-separability creates a situation in which the need for coordinated action
is clear to each firm in either case. Because of the complexity of selecting
any equilibrium set of production amounts Q, Runge (1981) does not
present a solution. Instead, he points out that the separable case creates a
prisoner's dilemma situation, and asserts that in the non-separable case an
assurance game results.

To make some headway for Runge's non-separable case, examine incentives to
defect from a jointly optimal solution, Q*, however that
solution was adopted. Suppose that through some mechanism (such as a series of
public discussions at village feasts), all firms are operating at a socially
optimal values
q1*,......qn*.What are the incentives to defect from this social optimum under, first, a
pure selfishness rule and, second, a symmetric generosity rule?

A defection by firm 1 would be an increase dq1 from
q1* at the social optimum, since all first partial
derivatives are positive,

it follows that in the selfish case, when vii= 1 and
vij = 0,

and an increase in q1* to
q1* + dq would increase the profit of the
first firm. There is a clear incentive to defect.

In the case of symmetric generosity, however, an increase in firm 1's profit is
matched by a decrease in the profit of all of the other firms. When
v11 = vij = 1/n, then there is no
incentive to defect, because the condition for a social optimum is that for the
first firm, price is equal to the sum of the marginal costs of all of the firms
with respect to the output of the first firm:

This same condition applies to all firms. The selected outputs, Q*, are
stable.